Animated Solution for Physics - Kinematics: In the cube of side '$a$' shown in the figure, the vector from the central point of the face $ABOD$ to the central point of the face $BEFO$ will be

## Finding the Vector Between Face Centers of a Cube

When dealing with 3D geometry and vectors, the most powerful tool in your arsenal is setting up a clean, logical coordinate system. By anchoring our geometry to the origin, complex spatial relationships reduce to simple algebraic subtractions.

Imagine a cube of side length $a$. To make our lives easy, let's place one of its vertices, $O$, exactly at the origin $(0,0,0)$. We align three of its edges along the positive $X$, $Y$, and $Z$ axes.

This means the vertices lying directly on the axes are: - $D=(a,0,0)$ on the $X$-axis. - $F=(0,a,0)$ on the $Y$-axis. - $B=(0,0,a)$ on the $Z$-axis.

The problem asks for the central point of the face $ABOD$. Looking at our setup, the vertices $O$, $D$, and $B$ lie in the $XZ$-plane. The fourth vertex $A$ must also lie in this plane, directly above $D$. Thus, $A$ has coordinates $(a,0,a)$.

Because $ABOD$ is a square, its geometric center (let's call it $G$) is simply the midpoint of its diagonal $OA$. Using the midpoint formula:

Next, we look at the face $BEFO$. The vertices $O$, $F$, and $B$ lie in the $YZ$-plane. The fourth vertex $E$ is directly above $F$, giving it the coordinates $(0,a,a)$.

Similarly, the center of this face (let's call it $H$) is the midpoint of the diagonal $OE$:

We need the vector from the center of $ABOD$ to the center of $BEFO$. This is the vector $\overrightarrow{GH}$. To find it, we subtract the position vector of the initial point $G$ from the final point $H$:

Substituting our coordinates:

Notice how the $Z$-components perfectly cancel out! This makes physical sense because both centers lie at the exact same height ($z=a/2$). The vector is purely horizontal.

Factoring out $\frac{a}{2}$, we get our final elegant result:

This perfectly matches option (b). By trusting the coordinate geometry, we bypassed any need for complicated 3D visualization and arrived straight at the answer.